Expectation value of two annihilation operators

  • #1

Main Question or Discussion Point

Hello,

I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following,
[tex] c = \frac{(a + ib)}{\sqrt{2}} [/tex]​

Now if I try to measure the intensity of this beam by a photodiode, the intensity will be proportional to-

[tex]<c^{\dag} c>[/tex]

On evaluating this, I get,
[tex]\frac{\left(<a^{\dag} a> + <b^{\dag} b> + i(<a^{\dag} b> - <b^{\dag} a>)\right)}{2} [/tex]
Now the book says, that this can be written as,
[tex]\frac{\left(<a^{\dag} a> + <b^{\dag} b> + i(<a^{\dag}><b> - <b^{\dag}><a>)\right)}{2} [/tex]

I am unable to understand this step, that is [tex]<a^{\dag} b> = <a^{\dag}><b>[/tex]
can someone please explain this.
I understand that these mode operators commute, but is this always true for any two commuting operators.

Thanks
 

Answers and Replies

  • #2
malawi_glenn
Science Advisor
Homework Helper
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if you write what [itex] < a^{\dagger} \, b > [/itex] is, then I am sure that you can figure it out.
 

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